We now consider the case where , i.e when F corresponds to
a complex cubic field. In this case the Hessian is an indefinite quadratic
form, and in general there will be many reduced quadratic forms equivalent
to it. Instead of using the Hessian, we will use a clever idea due to Matthews
and Berwick (see [9]). If , then F has a unique real root ,
and if F is irreducible, so if we factor F in as

the quadratic form will be definite ( i.e ), but with
real nonrational coefficients. We are going to show that the form
has many of the properties of the Hessian.

An easy computation gives

By changing to , i.e F into -F, we may assume that
, and if we have

hence

If , then a simple computation
shows that

the absolute value sign coming from the choice .

Definition 6.1 Let be an integral binary complex
cubic form, and let as above. We say that F is reduced
if 0<|Q|<P<R and if in addition a>0, , and d>0 if b=0.

Note that when F is irreducible, is irrational, hence the special
cases Q=0, P=|Q| or P=R which occurred in the real case cannot occur
here. Another nice fact is that we do not need to compute the irrational
numbers P, Q and R at all: